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Using their unitary extension principle, Ron and Shen have constructed a normalized tight frame for L2(R)L^2(\\mathbb {R}) consisting of spline functions with uniform knots. This paper constructs a normalized tight frame for L2((0,∞))L^2((0,\\infty )) comprising spline functions with knots on a hybrid of uniform and geometric mesh. The construction is motivated by applications in adaptive approximation using spline functions on a hybrid mesh that admits a natural dyadic multiresolution approximation of L2((0,∞))L^2((0,\\infty )) based on dilation and translation.

The Gaussian function$G(x) = \\frac{1}{\\sqrt{2\\pi}} e^{-x^2/2},$which has been a classical choice for multiscale representation, is the solution of the scaling equation \\[ G(x) = ınt_ R G( x - y) dg(y), xın R, \\] with scale$\\alpha >1$and absolutely continuous measure \\[ dg (y) = 12 (^2-1)e^-y^2/2(^2-1) dy. \\] It is known that the sequence of normalized B-splines (Bn), where Bn is the solution of the scaling equation$$ \\phi(x) = \\sum_{j=0}^n \\frac{1}{2^{n-1}} \\binom{n}{j} \\phi(2 x -j), \\quad xın {\\mathbb R}, $$converges uniformly to G. The classical results on normal approximation of binomial distributions and the uniform B-splines are studied in the broader context of normal approximation of probability measures mn, n=1,2,. . . , and the corresponding solutions$\\phi_n$of the scaling equations$ $_n (x) = ınt_ R _n ( x -y) dm_n (y), xın R.$$Various forms of convergence are considered and orders of convergence obtained. A class of probability densities are constructed that converge to the Gaussian function faster than the uniform B-splines.

We study the Cholesky factorization of certain bi-infinite matrices and related finite matrices. These results are applied to show that if the uniform translates of a suitably decaying multivariate function are orthonormalized by the Gram--Schmidt process over certain increasing finite sets, then the resulting functions converge to translates of a fixed function which is obtained by a global orthonormalization procedure. This convergence is also illustrated numerically.

Waiting at home are Miri's mother, Caroline (Geraldine James), and father, Oscar (Richard Durden), whose lives also have been derailed by her prison sentence. Haggard and Solon do an excellent job handling the nature of Miri's crime itself, which is revealed early on though remains the subject of some uncertainty when it comes to the exact details and circumstances. There are a number of excellent supporting turns, including Adeel Akhtar as a kind neighbor and Jo Martin as Miri's selfcentered parole officer.

Anne Hathaway, Tina Fey and Dev Patel lead a starry cast in Amazon's deft and winning series adaptation of the weekly New York Times column about relationships Sometimes a simple idea well executed is all you need. Carney deserves a lot of credit here; he wrote and directed half of the eight episodes and has guided the series toward a natural-feeling realism (even when the episode headlined by Hathaway dips into a dance number). In a standout episode, also directed by Carney, young developer Joshua (Patel) is interviewed about his online dating app by reporter Julie (Catherine Keener), revealing that his own chance at true love (with a woman played by Caitlin McGee) has been derailed.

In October 1989, a white man, Charles \"Chuck\" Stuart, murdered his wife, wounded himself and then claimed the killer was black; the Boston Police Department, in its investigation, resorted to intimidation and coercion in charging a black man; then Stuart's brother told the truth, lead ing to Stuart's suicide. For reasons that are not especially clear, both men will tackle the massive task of changing Boston's institutions - starting with getting criminals and everyday folks to rat out some bank robbers in Charlestown, the part of town where cases go to die. While that ostensibly is the main thrust of City on a Hill, these early episodes are mostly spent introducing supporting figures who surround Bacon and Hodge: bosses, siblings, mothers, friends, neighborhood bartenders and, of course, spouses (Jill Hennessy is a standout as Jackie's ignored wife) and colleagues in the DA's office (Sarah Shahi and Kevin Dunn are both potential breakouts).

Phoebe Waller-Bridge's tragicomic gem about a wayward Londoner returns to Amazon for a second season that doesn't shine quite as brightly By Tim Goodman Nearly three years after it premiered in the U.S. on Amazon, the critically acclaimed British series Fleabag, which launched its creator, writer and star Phoebe Waller-Bridge, returns for a second season. The inaugural season found WallerBridge workshopping her 2013 one-woman play of the same title to small-screen perfection; sophomore efforts often disappoint. By the end of season two's six half-hour episodes, there's a nagging sense that what we've just watched was the result of fans demanding a second act and Waller-Bridge, with some reluctance, saying, \"Sure, why not?\" And while it's a joy reuniting with these characters, there's less urgency to watching their lives play out.